A227135 Partitions with parts repeated at most twice and repetition only allowed if first part has an even index (first index = 1).

1, 1, 1, 2, 2, 4, 4, 6, 8, 10, 12, 17, 20, 25, 31, 39, 47, 58, 69, 85, 102, 123, 145, 175, 207, 246, 290, 343, 401, 473, 551, 646, 751, 875, 1012, 1177, 1358, 1570, 1807, 2083, 2389, 2746, 3140, 3597, 4106, 4690, 5337, 6082, 6907, 7848, 8895, 10085, 11404, 12902, 14561, 16438, 18520, 20864, 23460, 26385, 29619

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Formula

Conjecture: A227134(n) + A227135(n) = A182372(n) for n>=0, see comment in A182372.

G.f.: 1/(1-x) + Sum_{n>=2} x^(A002620(n+2)-1) / Product_{k=1..n} (1-x^k), where A002620(n) = floor(n/2)*ceiling(n/2) forms the quarter-squares. - Paul D. Hanna, Jul 06 2013

a(n) ~ c * exp(Pi*sqrt(2*n/5)) / n^(3/4), where c = 2^(3/4) / (sqrt(5)*(1 + sqrt(5))^(3/2)) = 0.1291995618069... - Vaclav Kotesovec, May 28 2018, updated Mar 06 2020

Example

G.f.: 1 + x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 4*x^6 + 6*x^7 + 8*x^8 +...
G.f.: 1/(1-x) + x^3/((1-x)*(1-x^2)) + x^5/((1-x)*(1-x^2)*(1-x^3)) + x^8/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) + x^11/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) + x^15/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6))) +...
There are a(13)=25 such partitions, displayed here as partitions into two sorts of parts (format P:S for sort:part) where the first sort is 1 and sorts oscillate:
01:  [ 1:1  2:0  2:1  3:0  5:1  ]
02:  [ 1:1  2:0  2:1  4:0  4:1  ]
03:  [ 1:1  2:0  2:1  8:0  ]
04:  [ 1:1  2:0  3:1  7:0  ]
05:  [ 1:1  2:0  4:1  6:0  ]
06:  [ 1:1  2:0 10:1  ]
07:  [ 1:1  3:0  3:1  6:0  ]
08:  [ 1:1  3:0  4:1  5:0  ]
09:  [ 1:1  3:0  9:1  ]
10:  [ 1:1  4:0  8:1  ]
11:  [ 1:1  5:0  7:1  ]
12:  [ 1:1  6:0  6:1  ]
13:  [ 1:1 12:0  ]
14:  [ 2:1  3:0  3:1  5:0  ]
15:  [ 2:1  3:0  8:1  ]
16:  [ 2:1  4:0  7:1  ]
17:  [ 2:1  5:0  6:1  ]
18:  [ 2:1 11:0  ]
19:  [ 3:1  4:0  6:1  ]
20:  [ 3:1  5:0  5:1  ]
21:  [ 3:1 10:0  ]
22:  [ 4:1  9:0  ]
23:  [ 5:1  8:0  ]
24:  [ 6:1  7:0  ]
25:  [13:1  ]

Maple

## See A227134
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, 1-t,
      `if`(i*(i+1)<n, 0, add(b(n-i*j, i-1,
      irem(t+j, 2)), j=0..min(t+1, n/i))))
    end:
a:= n-> add(b(n$2, t), t=0..1):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 15 2017

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1 - t, If[i*(i + 1) < n, 0, Sum[ b[n - i*j, i - 1, Mod[t + j, 2]], {j, 0, Min[t + 1, n/i]}]]];
a[n_] := Sum[b[n, n, t], {t, 0, 1}];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 21 2018, after Alois P. Heinz *)

Prog

(PARI) {A002620(n)=floor(n/2)*ceil(n/2)}
{a(n)=polcoeff(1/(1-x+x*O(x^n)) + sum(m=2, sqrtint(4*n), x^(A002620(m+2)-1)/prod(k=1, m, 1-x^k+x*O(x^n))), n)}
for(n=0, 60, print1(a(n), ", ")) \\ Paul D. Hanna, Jul 06 2013

Crossrefs

Cf. A227134 (parts may repeat after odd index).

Sequence in context: A057601 A294150 A087135 * A162417 A240012 A375134

Adjacent sequences: A227132 A227133 A227134 * A227136 A227137 A227138

Keyword

nonn

Author

Joerg Arndt, Jul 02 2013

Status

approved